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Proof of a conjecture on the total positivity of amazing matrices
Abstract
Let n and b be positive integers. Define the amazing matrix Pn,b=[P(i,j)]i,j=0n−1 to be an n×n matrix with entriesP(i,j)=1bn∑r≥0(−1)r(n+1r)(n−1−i+(j+1−r)bn). Diaconis and Fulman conjectured that the amazing matrix is totally positive. We give an affirmative answer to this conjecture.
... As a byproduct of our studies, we obtain in Theorem 3.7 additional structural insight into the combinatorics of H −1 F in this case. We recall results by Diaconis and Fulman [10] and Mao and Wang [11] which imply the H F is TP 2 [10] and even TP [11] if F is the rth-edgewise subdivision. ...
... As a byproduct of our studies, we obtain in Theorem 3.7 additional structural insight into the combinatorics of H −1 F in this case. We recall results by Diaconis and Fulman [10] and Mao and Wang [11] which imply the H F is TP 2 [10] and even TP [11] if F is the rth-edgewise subdivision. ...
... In case F is the rth-edgewise subdivision, an even stronger result holds. Theorem 2.5 (Mao and Wang [11]). Let F be the rth-edgewise subdivision. ...
Let be a -dimensional simplicial complex and its h-vector. For a face uniform subdivision operation , we write for the subdivided complex and for the matrix, such that . In connection with the real rootedness of symmetric decompositions, Athanasiadis and Tzanaki studied for strictly positive h-vectors the inequalities and . In this paper, we show that if the inequalities holds for a simplicial complex and is (all entries and two minors are non-negative), then the inequalities hold for . We prove that if is the barycentric subdivision, then is . If is the rth-edgewise subdivision, then work of Diaconis and Fulman shows is . Indeed, in this case by work of Mao and Wang, is even TP.
We prove the total positivity of the Narayana triangles of type A and type B, and thus affirmatively confirm a conjecture of Chen, Liang and Wang and a conjecture of Pan and Zeng. We also prove the strict total positivity of the Narayana squares of type A and type B.
We show that, for Hankel matrices, total nonnegativity (resp. total positivity) of order r is preserved by sum, Hadamard product, and Hadamard power with real exponent t \ge r-2. We give examples to show that our results are sharp relative to matrix size and structure (general, symmetric or Hankel). Some of these examples also resolve the Hadamard critical-exponent problem for totally positive and totally nonnegative matrices.
Given a polynomial with positive
coefficients , and a positive integer , we define a(n infinite)
generalized Hurwitz matrix . We prove that the
polynomial f(z) does not vanish in the sector whenever the matrix is totally
nonnegative. This result generalizes the classical Hurwitz' Theorem on stable
polynomials (M=2), the Aissen-Edrei-Schoenberg-Whitney theorem on polynomials
with negative real roots (M=1), and the Cowling-Thron theorem (M=n). In
this connection, we also develop a generalization of the classical Euclidean
algorithm, of independent interest per se.
We give a detailed account of various connections between several classes of
objects: Hankel, Hurwitz, Toeplitz, Vandermonde and other structured matrices,
Stietjes and Jacobi-type continued fractions, Cauchy indices, moment problems,
total positivity, and root localization of univariate polynomials. Along with a
survey of many classical facts, we provide a number of new results.
Let R=R(d(t),h(t)) be a Riordan array, where d(t)=∑n≥0dntn and h(t)=∑n≥0hntn. We show that if the matrix[d0h000⋯d1h1h00d2h2h1h0⋮⋮⋱] is totally positive, then so is the Riordan array R.
Many combinatorial matrices — such as those of binomial coefficients, Stirling numbers of both kinds, and Lah numbers — are known to be totally non-negative, meaning that all minors (determinants of square submatrices) are non-negative.
The examples noted above can be placed in a common framework: for each one there is a non-decreasing sequence (a1,a2,…), and a sequence (e1,e2,…), such that the (m,k) entry of the matrix is the coefficient of the polynomial (x−a1)⋯(x−ak) in the expansion of (x−e1)⋯(x−em) as a linear combination of the polynomials 1,x−a1,…,(x−a1)⋯(x−am).
We consider this general framework. For a non-decreasing sequence (a1,a2,…) we establish necessary and sufficient conditions on the sequence (e1,e2,…) for the corresponding matrix to be totally non-negative. As corollaries we obtain total non-negativity of matrices of rook numbers of Ferrers boards, and of graph Stirling numbers of chordal graphs.
The Delannoy numbers d(n,k) count the number of lattice paths from (0,0) to (n−k,k) using steps (1,0),(0,1) and (1,1). We show that the zeros of all Delannoy polynomials d n (x)=∑ k=0ⁿ d(n,k)x k are in the open interval (−3−22,−3+22) and are dense in the corresponding closed interval. We also show that the Delannoy numbers d(n,k) are asymptotically normal (by central and local limit theorems).
Let R=(d(t),h(t)) be a Riordan array. We show that if both d(t) and h(t) are Pólya frequency formal power series, then R is totally positive.
Let be an infinite lower triangular matrix defined
by the recurrence where
unless and are all nonnegative. Many well-known
combinatorial triangles are such matrices, including the Pascal triangle, the
Stirling triangle (of the second kind), the Bell triangle, the Catalan
triangles of Aigner and Shapiro. We present some sufficient conditions such
that the recursive matrix A is totally positive. As applications we give the
total positivity of the above mentioned combinatorial triangles in a unified
approach.
We show that the Catalan triangle B = [B-n,B-k](n,k >= 1), defined by B-n,B-k = k/n ((n + k) (2n)), is a totally positive matrix.
We present sufficient conditions for total positivity of Riordan arrays. As
applications we show that many well-known combinatorial triangles are totally
positive and many famous combinatorial numbers are log-convex in a unified
approach.
Total positivity arises often in various branches of mathematics, statistics, probability, mechanics, economics, and computer science (see, e.g., [24], and the references cited there). In this paper we give a survey of the interactions between total positivity and combinatorics.
We study a generalization of Holteʼs amazing matrix, the transition probability matrix of the Markov chains of the ‘carries’ in a non-standard numeration system. The stationary distributions are explicitly described by the numbers which can be regarded as a generalization of the Eulerian numbers and the MacMahon numbers. We also show that similar properties hold even for the numeration systems with the negative bases.
We study the combinatorics of addition using balanced digits, deriving an
analog of Holte's "amazing matrix" for carries in usual addition. The
eigenvalues of this matrix for base b balanced addition of n numbers are found
to be 1,1/b,...,1/b^n, and formulas are given for its left and right
eigenvectors. It is shown that the left eigenvectors can be identified with
hyperoctahedral Foulkes characters, and that the right eigenvectors can be
identified with hyperoctahedral Eulerian idempotents. We also examine the
carries that occur when a column of balanced digits is added, showing this
process to be determinantal. The transfer matrix method and a serendipitous
diagonalization are used to study this determinantal process.
The “carries” when n random numbers are added base b form a Markov chain with an “amazing” transition matrix determined in a 1997 paper of Holte. This same Markov chain occurs in following the number of descents when n cards are repeatedly riffle shuffled. We give generating and symmetric function proofs and determine the rate of convergence of this Markov chain to stationarity. Similar results are given for type B shuffles. We also develop connections with Gaussian autoregressive processes and the Veronese mapping of commutative algebra.
It was first observed in (F. Brenti, Mem. Amer. Math. Soc.413, 1989) that Pólya frequency sequences arise often in combinatorics. In this work we point out that the same is true, more generally, for totally positive matrices and that there is an intimate connection between them and some generalizations of the classical symmetric functions.
Let (an)n⩾0 be a sequence of complex numbers such that its generating series satisfies for some polynomial h(t). For any r⩾1 we study the transformation of the coefficient series of h(t) to that of h〈r〉(t) where . We give a precise description of this transformation and show that under some natural mild hypotheses the roots of h〈r〉(t) converge when r goes to infinity. In particular, this holds if ∑n⩾0antn is the Hilbert series of a standard graded k-algebra A. If in addition A is Cohen–Macaulay then the coefficients of h〈r〉(t) are monotonically increasing with r. If A is the Stanley–Reisner ring of a simplicial complex Δ then this relates to the rth edgewise subdivision of Δ—a subdivision operation relevant in computational geometry and graphics—which in turn allows some corollaries on the behavior of the respective f-vectors.
We show that a refinable function φ with dilation M⩾2 is a ripplet, i.e., the collocation matrices of its shifts are totally positive, provided that the symbol p of its refinement mask satisfies certain conditions. The main condition is that p (of degree n) satisfies what we term condition (I), which requires that n determinants of the coefficients of p are positive and generalises the conditions of Hurwitz for a polynomial to have all negative zeros. We also generalise a result of Kemperman to show that (I) is equivalent to an M-slanted matrix of the coefficients of p being totally positive. Under condition (I), the ripplet φ satisfies a generalisation of the Schoenberg–Whitney conditions provided that n is an integer multiple of M−1. Moreover, (I) implies that polynomials in a polyphase decomposition of p have interlacing negative zeros, and under these weaker conditions we show that φ still enjoys certain total positivity properties.
We present a simple way to derive the results of Diaconis and Fulman
[arXiv:1102.5159] in terms of noncommutative symmetric functions.
John Holte [16] introduced a family of "amazing matrices" which give the
transition probabilities of "carries" when adding a list of numbers. It was
subsequently shown that these same matrices arise in the combinatorics of the
Veronese embedding of commutative algebra [4,6,7] and in the analysis of riffle
shuffling [6,7]. We find that the left eigenvectors of these matrices form the
Foulkes character table of the symmetric group and the right eigenvectors are
the Eulerian idempotents introduced by Loday [20] in work on Hochschild
homology. The connections give new closed formulae for Foulkes characters and
allow explicit computation of natural correlation functions in the original
carries problem.
The number of ``carries'' when n random integers are added forms a Markov chain [23]. We show that this Markov chain has the same transition matrix as the descent process when a deck of n cards is repeatedly riffle shuffled. This gives new results for the statistics of carries and shuffling.